Contents
sstegr - (a) Compute T-sigma_i = L_i D_i L_i^T, such that
L_i D_i L_i^T is a relatively robust representation
SUBROUTINE SSTEGR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER ISUPPZ(*), IWORK(*)
REAL VL, VU, ABSTOL
REAL D(*), E(*), W(*), Z(LDZ,*), WORK(*)
SUBROUTINE SSTEGR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M,
W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)
CHARACTER * 1 JOBZ, RANGE
INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER*8 ISUPPZ(*), IWORK(*)
REAL VL, VU, ABSTOL
REAL D(*), E(*), W(*), Z(LDZ,*), WORK(*)
F95 INTERFACE
SUBROUTINE STEGR(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, ABSTOL, M,
W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE
INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: ISUPPZ, IWORK
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: D, E, W, WORK
REAL, DIMENSION(:,:) :: Z
SUBROUTINE STEGR_64(JOBZ, RANGE, [N], D, E, VL, VU, IL, IU, ABSTOL,
M, W, Z, [LDZ], ISUPPZ, [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE
INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK
REAL :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: D, E, W, WORK
REAL, DIMENSION(:,:) :: Z
C INTERFACE
#include <sunperf.h>
void sstegr(char jobz, char range, int n, float *d, float
*e, float vl, float vu, int il, int iu, float
abstol, int *m, float *w, float *z, int ldz, int
*isuppz, int *info);
void sstegr_64(char jobz, char range, long n, float *d,
float *e, float vl, float vu, long il, long iu,
float abstol, long *m, float *w, float *z, long
ldz, long *isuppz, long *info);
sstegr b) Compute the eigenvalues, lambda_j, of L_i D_i
L_i^T to high
relative accuracy by the dqds algorithm,
(c) If there is a cluster of close eigenvalues, "choose"
sigma_i
close to the cluster, and go to step (a),
(d) Given the approximate eigenvalue lambda_j of L_i D_i
L_i^T,
compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the
input parameter ABSTOL.
For more details, see "A new O(n^2) algorithm for the sym-
metric tridiagonal eigenvalue/eigenvector problem", by
Inderjit Dhillon, Computer Science Division Technical Report
No. UCB/CSD-97-971, UC Berkeley, May 1997.
Note 1 : Currently SSTEGR is only set up to find ALL the n
eigenvalues and eigenvectors of T in O(n^2) time
Note 2 : Currently the routine SSTEIN is called when an
appropriate sigma_i cannot be chosen in step (c) above.
SSTEIN invokes modified Gram-Schmidt when eigenvalues are
close.
Note 3 : SSTEGR works only on machines which follow ieee-754
floating-point standard in their handling of infinities and
NaNs. Normal execution of SSTEGR may create NaNs and infin-
ities and hence may abort due to a floating point exception
in environments which do not conform to the ieee standard.
JOBZ (input)
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
RANGE (input)
= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval
(VL,VU] will be found. = 'I': the IL-th through
IU-th eigenvalues will be found.
N (input) The order of the matrix. N >= 0.
D (input/output)
On entry, the n diagonal elements of the tridiago-
nal matrix T. On exit, D is overwritten.
E (input/output)
On entry, the (n-1) subdiagonal elements of the
tridiagonal matrix T in elements 1 to N-1 of E;
E(N) need not be set. On exit, E is overwritten.
VL (input)
If RANGE='V', the lower and upper bounds of the
interval to be searched for eigenvalues. VL < VU.
Not referenced if RANGE = 'A' or 'I'.
VU (input)
See the description of VL.
IL (input)
If RANGE='I', the indices (in ascending order) of
the smallest and largest eigenvalues to be
returned. 1 <= IL <= IU <= N, if N > 0; IL = 1
and IU = 0 if N = 0. Not referenced if RANGE =
'A' or 'V'.
IU (input)
See the description of IL.
ABSTOL (input)
The absolute error tolerance for the
eigenvalues/eigenvectors. IF JOBZ = 'V', the
eigenvalues and eigenvectors output have residual
norms bounded by ABSTOL, and the dot products
between different eigenvectors are bounded by
ABSTOL. If ABSTOL is less than N*EPS*|T|, then
N*EPS*|T| will be used in its place, where EPS is
the machine precision and |T| is the 1-norm of the
tridiagonal matrix. The eigenvalues are computed
to an accuracy of EPS*|T| irrespective of ABSTOL.
If high relative accuracy is important, set ABSTOL
to DLAMCH( 'Safe minimum' ). See Barlow and Dem-
mel "Computing Accurate Eigensystems of Scaled
Diagonally Dominant Matrices", LAPACK Working Note
#7 for a discussion of which matrices define their
eigenvalues to high relative accuracy.
M (output)
The total number of eigenvalues found. 0 <= M <=
N. If RANGE = 'A', M = N, and if RANGE = 'I', M =
IU-IL+1.
W (output)
The first M elements contain the selected eigen-
values in ascending order.
Z (input) If JOBZ = 'V', then if INFO = 0, the first M
columns of Z contain the orthonormal eigenvectors
of the matrix T corresponding to the selected
eigenvalues, with the i-th column of Z holding the
eigenvector associated with W(i). If JOBZ = 'N',
then Z is not referenced. Note: the user must
ensure that at least max(1,M) columns are supplied
in the array Z; if RANGE = 'V', the exact value of
M is not known in advance and an upper bound must
be used.
LDZ (input)
The leading dimension of the array Z. LDZ >= 1,
and if JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ (output)
The support of the eigenvectors in Z, i.e., the
indices indicating the nonzero elements in Z. The
i-th eigenvector is nonzero only in elements
ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
(and minimal) LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >=
max(1,18*N)
If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the WORK array, returns this value as the first
entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.
IWORK (workspace/output)
On exit, if INFO = 0, IWORK(1) returns the optimal
LIWORK.
LIWORK (input)
The dimension of the array IWORK. LIWORK >=
max(1,10*N)
If LIWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the IWORK array, returns this value as the first
entry of the IWORK array, and no error message
related to LIWORK is issued by XERBLA.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value
> 0: if INFO = 1, internal error in SLARRE, if
INFO = 2, internal error in SLARRV.
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA